The phenomenon of “missing mass” in galaxies has triggered new theoretical exploration, forming a competition between dark matter assumption, modified Newtonian dynamics and modified gravity. Over the past forty years, various versions of the modified scenario have been proposed to simulate the effects of missing mass. These schemes replace the dynamic effect of dark matter by introducing some tiny extra force terms in the dynamic equations. Such extra forces have mainly interactions on large scales of galaxies, such as fitting the Tully-Fisher relation or asymptotically flat rotation curves. The discussion in this paper shows that the evidence of taking the modified schemes as fundamental theory is still insufficient. In this paper, we display a system of simplified galactic dynamical equations derived from weak field and low-speed approximations of Einstein field equations, and then we use it to discuss two important empirical relations in galactic dynamics, namely the Faber-Jackson relation and Tully-Fisher relation, as well as the related fundamental plane. These discussions provide a reference scheme for improving the dispersion of the empirical relations, and also provide a theoretical foundation to analyze the properties of dark matter and galactic structures.
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